Censoring and Stochastic Integrals

Suppose Y is a real-valued function on IR which is right continuous with left hand limits and is of bounded variation on each bounded subinterval of IR+ (we also say "Y is of locally bounded variation"). Moreover suppose that X is a Lebesgue-measurable real-valued function on JR.+ such that JsdO,t]!X(s) I ldY(s) I is finite for each t E IR+ (i.e. "X is locally integrable with respect to Y"). Here the integral is a Lebesgue-Stieltjes integral with respect to the total variation of Y (which assigns mass !Y(Oll to the point zero in line with the convention Y(O-) = 0). Then for each t we define